• Previous Article
    Approximation of linear controlled dynamical systems with small random noise and fast periodic sampling
  • MCRF Home
  • This Issue
  • Next Article
    Theoretical and computational decay results for a memory type wave equation with variable-exponent nonlinearity
doi: 10.3934/mcrf.2022006
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

The discretized backstepping method: An application to a general system of $ 2\times 2 $ linear balance laws

Institut de mathematiques de Toulouse, 118 route de Narbonne, Toulouse, France

Received  July 2021 Revised  January 2022 Early access March 2022

In this paper, we introduce the numerical backstepping method by applying it to a problem of finite-time stabilization for a system of $ 2 \times 2 $ balance laws discretized thanks to the upwind scheme. On the one hand, we illustrate on an example that the scheme used to compute the feedback control cannot be chosen arbitrarily. On the other hand, an algorithm is given to construct this control properly and an approached finite-time stabilization result is proven.

Citation: Mathias Dus. The discretized backstepping method: An application to a general system of $ 2\times 2 $ linear balance laws. Mathematical Control and Related Fields, doi: 10.3934/mcrf.2022006
References:
[1]

J. Auriol, Robust Design of Backstepping Controllers for Systems of Linearhyperbolic PDEs, PhD thesis, Mines ParisTech, Paris, 2018.

[2]

A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-938.  doi: 10.1137/S0036139997332099.

[3]

G. Bastin and J.-M. Coron, On boundary feedback stabilization of non-uniform linear $2\times2$ hyperbolic systems over a bounded interval, Systems Control Lett., 60 (2011), 900-906.  doi: 10.1016/j.sysconle.2011.07.008.

[4]

G. Bastin and J.-M. Coron, Stability and Boundary Stabilization of 1-D Hyperbolic Systems, Progress in Nonlinear Differential Equations and their Applications, 88. Subseries in Control. Birkhäuser/Springer, [Cham], 2016. doi: 10.1007/978-3-319-32062-5.

[5]

G. BastinJ.-M. Coron and B. d'Andréa Novel, On Lyapunov stability of linearised Saint-Venant equations for a sloping channel, Netw. Heterog. Media, 4 (2009), 177-187.  doi: 10.3934/nhm.2009.4.177.

[6]

J.-M. Coron, On the null asymptotic stabilization of the two-dimensional incompressible Euler equations in a simply connected domain, SIAM J. Control Optim., 37 (1999), 1874-1896.  doi: 10.1137/S036301299834140X.

[7]

J.-M. CoronR. VazquezM. Krstic and G. Bastin, Local exponential $H^2$ stabilization of a $2\times 2$ quasilinear hyperbolic system using backstepping, SIAM J. Control Optim., 51 (2013), 2005-2035.  doi: 10.1137/120875739.

[8]

J. de HalleuxC. PrieurJ.-M. CoronB. Novel and G. Bastin, Boundary feedback control in networks of open channels, Automatica J. IFAC, 39 (2003), 1365-1376.  doi: 10.1016/S0005-1098(03)00109-2.

[9]

F. Di Meglio, Dynamic and Control of Slugging in Oil Production, PhD thesis, Mines ParisTech, Paris, 2011.

[10]

F. Di Meglio, G. Kaasa, N. Petit and V. Alstad, Slugging in multiphase flow as a mixed initial-boundary value problem for a quasilinear hyperbolic system, In Proceedings of the 2011 American Control Conference, (2011), 3589–3596.

[11]

F. Di MeglioR. Vazquez and M. Krstic, Stabilization of a system of $n+1$ coupled first-order hyperbolic linear pdes with a single boundary input, IEEE Trans. Automat. Control, 58 (2013), 3097-3111.  doi: 10.1109/TAC.2013.2274723.

[12]

A. DiagneG. Bastin and J.-M. Coron, Lyapunov exponential stability of 1-D linear hyperbolic systems of balance laws, Automatica J. IFAC, 48 (2012), 109-114.  doi: 10.1016/j.automatica.2011.09.030.

[13]

S. Dudret, K. Beauchard, F. Ammouri and P. Rouchon, Stability and asymptotic observers of binary distillation processes described by nonlinear convection/diffusion models, In 2012 American Control Conference (ACC), (2012), 3352–3358.

[14]

M. Dus, Exponential stability of a general slope limiter scheme for scalar conservation laws subject to a dissipative boundary condition, Mathematics of Control, Signals, and Systems, 34 (2022), 37-65.  doi: 10.1007/s00498-021-00301-2.

[15]

R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, In Handb. Numer. Anal., (2000), 713–1020.

[16]

A. Hayat, Boundary stability of 1-D nonlinear inhomogeneous hyperbolic systems for the $C^1$ norm, SIAM J. Control Optim., 57 (2019), 3603-3638.  doi: 10.1137/17M1150803.

[17]

A. Hayat, On boundary stability of inhomogeneous $2\times2$ 1-D hyperbolic systems for the $C^1$ norm, ESAIM Control Optim. Calc. Var., 25 (2019), Paper No. 82, 31 pp. doi: 10.1051/cocv/2018059.

[18]

J. M. Holte, Discrete Gronwall Lemma and Applications, Technical report, MAA north central section meeting at und, 2009.

[19]

L. HuF. Di MeglioR. Vazquez and M. Krstic, Control of homodirectional and general heterodirectional linear coupled hyperbolic PDEs, IEEE Trans. Automat. Control, 61 (2016), 3301-3314.  doi: 10.1109/TAC.2015.2512847.

[20]

L. Hu and G. Olive, Null controllability and finite-time stabilization in minimal time of one-dimensional first-order 2 $\times$ 2 linear hyperbolic systems, ESAIM Control Optim. Calc. Var., 27 (2021), Paper No. 96, 18 pp. doi: 10.1051/cocv/2021091.

[21]

M. Krstic, P. V. Kokotovic and I. Kanellakopoulos, Nonlinear and Adaptive Control Design, 1$^{st}$ edition, John Wiley and Sons, Inc., USA, 1995.

[22]

M. Krstic and A. Smyshlyaev, Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays, Systems Control Lett., 57 (2008), 750-758.  doi: 10.1016/j.sysconle.2008.02.005.

[23]

M. Krstic and A. Smyshlyaev, Boundary Control of PDEs, A course on backstepping designs. Advances in Design and Control, 16. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008. doi: 10.1137/1.9780898718607.

[24] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2002.  doi: 10.1017/CBO9780511791253.
[25]

A. SmyshlyaevB. Guo and M. Krstic, Arbitrary decay rate for euler-bernoulli beam by backstepping boundary feedback, IEEE Trans. Automat. Control, 54 (2009), 1134-1140.  doi: 10.1109/TAC.2009.2013038.

[26] A. Smyshlyaev and M. Krstic, Adaptive Control of Parabolic PDEs, Princeton University Press, Princeton, NJ, 2010.  doi: 10.1515/9781400835362.
[27]

R. Vazquez and M. Krstic, Marcum $Q$-functions and explicit kernels for stabilization of $2\times2$ linear hyperbolic systems with constant coefficients, Systems Control Lett., 68 (2014), 33-42.  doi: 10.1016/j.sysconle.2014.02.008.

[28]

C.-Z. Xu and G. Sallet, Exponential stability and transfer functions of processes governed by symmetric hyperbolic systems, ESAIM Control Optim. Calc. Var., 7 (2002), 421-442.  doi: 10.1051/cocv:2002062.

show all references

References:
[1]

J. Auriol, Robust Design of Backstepping Controllers for Systems of Linearhyperbolic PDEs, PhD thesis, Mines ParisTech, Paris, 2018.

[2]

A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-938.  doi: 10.1137/S0036139997332099.

[3]

G. Bastin and J.-M. Coron, On boundary feedback stabilization of non-uniform linear $2\times2$ hyperbolic systems over a bounded interval, Systems Control Lett., 60 (2011), 900-906.  doi: 10.1016/j.sysconle.2011.07.008.

[4]

G. Bastin and J.-M. Coron, Stability and Boundary Stabilization of 1-D Hyperbolic Systems, Progress in Nonlinear Differential Equations and their Applications, 88. Subseries in Control. Birkhäuser/Springer, [Cham], 2016. doi: 10.1007/978-3-319-32062-5.

[5]

G. BastinJ.-M. Coron and B. d'Andréa Novel, On Lyapunov stability of linearised Saint-Venant equations for a sloping channel, Netw. Heterog. Media, 4 (2009), 177-187.  doi: 10.3934/nhm.2009.4.177.

[6]

J.-M. Coron, On the null asymptotic stabilization of the two-dimensional incompressible Euler equations in a simply connected domain, SIAM J. Control Optim., 37 (1999), 1874-1896.  doi: 10.1137/S036301299834140X.

[7]

J.-M. CoronR. VazquezM. Krstic and G. Bastin, Local exponential $H^2$ stabilization of a $2\times 2$ quasilinear hyperbolic system using backstepping, SIAM J. Control Optim., 51 (2013), 2005-2035.  doi: 10.1137/120875739.

[8]

J. de HalleuxC. PrieurJ.-M. CoronB. Novel and G. Bastin, Boundary feedback control in networks of open channels, Automatica J. IFAC, 39 (2003), 1365-1376.  doi: 10.1016/S0005-1098(03)00109-2.

[9]

F. Di Meglio, Dynamic and Control of Slugging in Oil Production, PhD thesis, Mines ParisTech, Paris, 2011.

[10]

F. Di Meglio, G. Kaasa, N. Petit and V. Alstad, Slugging in multiphase flow as a mixed initial-boundary value problem for a quasilinear hyperbolic system, In Proceedings of the 2011 American Control Conference, (2011), 3589–3596.

[11]

F. Di MeglioR. Vazquez and M. Krstic, Stabilization of a system of $n+1$ coupled first-order hyperbolic linear pdes with a single boundary input, IEEE Trans. Automat. Control, 58 (2013), 3097-3111.  doi: 10.1109/TAC.2013.2274723.

[12]

A. DiagneG. Bastin and J.-M. Coron, Lyapunov exponential stability of 1-D linear hyperbolic systems of balance laws, Automatica J. IFAC, 48 (2012), 109-114.  doi: 10.1016/j.automatica.2011.09.030.

[13]

S. Dudret, K. Beauchard, F. Ammouri and P. Rouchon, Stability and asymptotic observers of binary distillation processes described by nonlinear convection/diffusion models, In 2012 American Control Conference (ACC), (2012), 3352–3358.

[14]

M. Dus, Exponential stability of a general slope limiter scheme for scalar conservation laws subject to a dissipative boundary condition, Mathematics of Control, Signals, and Systems, 34 (2022), 37-65.  doi: 10.1007/s00498-021-00301-2.

[15]

R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, In Handb. Numer. Anal., (2000), 713–1020.

[16]

A. Hayat, Boundary stability of 1-D nonlinear inhomogeneous hyperbolic systems for the $C^1$ norm, SIAM J. Control Optim., 57 (2019), 3603-3638.  doi: 10.1137/17M1150803.

[17]

A. Hayat, On boundary stability of inhomogeneous $2\times2$ 1-D hyperbolic systems for the $C^1$ norm, ESAIM Control Optim. Calc. Var., 25 (2019), Paper No. 82, 31 pp. doi: 10.1051/cocv/2018059.

[18]

J. M. Holte, Discrete Gronwall Lemma and Applications, Technical report, MAA north central section meeting at und, 2009.

[19]

L. HuF. Di MeglioR. Vazquez and M. Krstic, Control of homodirectional and general heterodirectional linear coupled hyperbolic PDEs, IEEE Trans. Automat. Control, 61 (2016), 3301-3314.  doi: 10.1109/TAC.2015.2512847.

[20]

L. Hu and G. Olive, Null controllability and finite-time stabilization in minimal time of one-dimensional first-order 2 $\times$ 2 linear hyperbolic systems, ESAIM Control Optim. Calc. Var., 27 (2021), Paper No. 96, 18 pp. doi: 10.1051/cocv/2021091.

[21]

M. Krstic, P. V. Kokotovic and I. Kanellakopoulos, Nonlinear and Adaptive Control Design, 1$^{st}$ edition, John Wiley and Sons, Inc., USA, 1995.

[22]

M. Krstic and A. Smyshlyaev, Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays, Systems Control Lett., 57 (2008), 750-758.  doi: 10.1016/j.sysconle.2008.02.005.

[23]

M. Krstic and A. Smyshlyaev, Boundary Control of PDEs, A course on backstepping designs. Advances in Design and Control, 16. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008. doi: 10.1137/1.9780898718607.

[24] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2002.  doi: 10.1017/CBO9780511791253.
[25]

A. SmyshlyaevB. Guo and M. Krstic, Arbitrary decay rate for euler-bernoulli beam by backstepping boundary feedback, IEEE Trans. Automat. Control, 54 (2009), 1134-1140.  doi: 10.1109/TAC.2009.2013038.

[26] A. Smyshlyaev and M. Krstic, Adaptive Control of Parabolic PDEs, Princeton University Press, Princeton, NJ, 2010.  doi: 10.1515/9781400835362.
[27]

R. Vazquez and M. Krstic, Marcum $Q$-functions and explicit kernels for stabilization of $2\times2$ linear hyperbolic systems with constant coefficients, Systems Control Lett., 68 (2014), 33-42.  doi: 10.1016/j.sysconle.2014.02.008.

[28]

C.-Z. Xu and G. Sallet, Exponential stability and transfer functions of processes governed by symmetric hyperbolic systems, ESAIM Control Optim. Calc. Var., 7 (2002), 421-442.  doi: 10.1051/cocv:2002062.

Figure 1.  The domain where $ P $ is defined
Figure 2.  The grid for the computation of $ P $
Figure 3.  The $ L^2 $ norm of the solution for case 1
Figure 4.  Spectrums of discretized operators for case 1
Figure 5.  The $ L^2 $ norm of the solution for case 2
Figure 6.  Spectra of discretized operators for case 2
Figure 7.  The space grids for $ \alpha = 2 $
Figure 8.  The definition of $ \Pi_{f \leftarrow c} $
Figure 9.  The definition of $ \Pi_{c \leftarrow f} $
Figure 10.  The non zero coefficients for $ \Gamma_{11} $ ($ \alpha = 3 $)
Figure 11.  The non zero coefficients for $ \Gamma_{12} $ ($ \alpha = 3 $)
Figure 12.  The kernels of the closed-loop operator for the naive method
Figure 13.  The spectrum of the closed-loop operator for the naive method
Figure 14.  The kernels of the closed-loop operator for less naive method
Figure 15.  The spectrum of the closed-loop operator for the less naive method
Figure 16.  The $ L^2 $ norm of the solution (log10 scale)
Figure 17.  When the perturbation term is non zero
Figure 18.  When the perturbation term is zero
Figure 19.  When velocities are different
[1]

Fatiha Alabau-Boussouira, Vincent Perrollaz, Lionel Rosier. Finite-time stabilization of a network of strings. Mathematical Control and Related Fields, 2015, 5 (4) : 721-742. doi: 10.3934/mcrf.2015.5.721

[2]

Arno Berger. On finite-time hyperbolicity. Communications on Pure and Applied Analysis, 2011, 10 (3) : 963-981. doi: 10.3934/cpaa.2011.10.963

[3]

Juanjuan Huang, Yan Zhou, Xuerong Shi, Zuolei Wang. A single finite-time synchronization scheme of time-delay chaotic system with external periodic disturbance. Mathematical Foundations of Computing, 2019, 2 (4) : 333-346. doi: 10.3934/mfc.2019021

[4]

Ta T.H. Trang, Vu N. Phat, Adly Samir. Finite-time stabilization and $H_\infty$ control of nonlinear delay systems via output feedback. Journal of Industrial and Management Optimization, 2016, 12 (1) : 303-315. doi: 10.3934/jimo.2016.12.303

[5]

Arno Berger, Doan Thai Son, Stefan Siegmund. Nonautonomous finite-time dynamics. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 463-492. doi: 10.3934/dcdsb.2008.9.463

[6]

Sanjeeva Balasuriya. Uncertainty in finite-time Lyapunov exponent computations. Journal of Computational Dynamics, 2020, 7 (2) : 313-337. doi: 10.3934/jcd.2020013

[7]

Jianjun Paul Tian. Finite-time perturbations of dynamical systems and applications to tumor therapy. Discrete and Continuous Dynamical Systems - B, 2009, 12 (2) : 469-479. doi: 10.3934/dcdsb.2009.12.469

[8]

Shu Dai, Dong Li, Kun Zhao. Finite-time quenching of competing species with constrained boundary evaporation. Discrete and Continuous Dynamical Systems - B, 2013, 18 (5) : 1275-1290. doi: 10.3934/dcdsb.2013.18.1275

[9]

Grzegorz Karch, Kanako Suzuki, Jacek Zienkiewicz. Finite-time blowup of solutions to some activator-inhibitor systems. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 4997-5010. doi: 10.3934/dcds.2016016

[10]

Thierry Cazenave, Yvan Martel, Lifeng Zhao. Finite-time blowup for a Schrödinger equation with nonlinear source term. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 1171-1183. doi: 10.3934/dcds.2019050

[11]

Emilija Bernackaitė, Jonas Šiaulys. The finite-time ruin probability for an inhomogeneous renewal risk model. Journal of Industrial and Management Optimization, 2017, 13 (1) : 207-222. doi: 10.3934/jimo.2016012

[12]

Tingting Su, Xinsong Yang. Finite-time synchronization of competitive neural networks with mixed delays. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3655-3667. doi: 10.3934/dcdsb.2016115

[13]

Tianhu Yu, Jinde Cao, Chuangxia Huang. Finite-time cluster synchronization of coupled dynamical systems with impulsive effects. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3595-3620. doi: 10.3934/dcdsb.2020248

[14]

Khalid Addi, Samir Adly, Hassan Saoud. Finite-time Lyapunov stability analysis of evolution variational inequalities. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1023-1038. doi: 10.3934/dcds.2011.31.1023

[15]

Peter Giesl. Construction of a finite-time Lyapunov function by meshless collocation. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2387-2412. doi: 10.3934/dcdsb.2012.17.2387

[16]

Gang Tian. Finite-time singularity of Kähler-Ricci flow. Discrete and Continuous Dynamical Systems, 2010, 28 (3) : 1137-1150. doi: 10.3934/dcds.2010.28.1137

[17]

Huijuan Li, Junxia Wang. Input-to-state stability of continuous-time systems via finite-time Lyapunov functions. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 841-857. doi: 10.3934/dcdsb.2019192

[18]

Zengyun Wang, Jinde Cao, Zuowei Cai, Lihong Huang. Finite-time stability of impulsive differential inclusion: Applications to discontinuous impulsive neural networks. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2677-2692. doi: 10.3934/dcdsb.2020200

[19]

Rui Li, Yingjing Shi. Finite-time optimal consensus control for second-order multi-agent systems. Journal of Industrial and Management Optimization, 2014, 10 (3) : 929-943. doi: 10.3934/jimo.2014.10.929

[20]

M. Syed Ali, L. Palanisamy, Nallappan Gunasekaran, Ahmed Alsaedi, Bashir Ahmad. Finite-time exponential synchronization of reaction-diffusion delayed complex-dynamical networks. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1465-1477. doi: 10.3934/dcdss.2020395

2021 Impact Factor: 1.141

Article outline

Figures and Tables

[Back to Top]